Markov bridges: SDE representation

Abstract

Let $X$ be a Markov process taking values in $\mathbf{E}$ with continuous paths and transition function $(P_{s,t})$. Given a measure $\mu$ on $(\mathbf{E},\mathcal{E})$, a Markov bridge starting at $(s,{\epsilon }_x)$ and ending at $(T^{\ast },\mu )$ for $T^{\ast }<\infty$ has the law of the original process starting at $x$ at time $s$ and conditioned to have law $\mu$ at time $T^{\ast }$. We will consider two types of conditioning: a) {\em weak conditioning} when $\mu$ is absolutely continuous with respect to $P_{s,t}(x,\cdot )$ and b) {\em strong conditioning} when $\mu ={\epsilon }_z$ for some $z\in \mathbf{E}$. The main result of this paper is the representation of a Markov bridge as a solution to a stochastic differential equation (SDE) driven by a Brownian motion in a diffusion setting. Under mild conditions on the transition density of the underlying diffusion process we establish the existence and uniqueness of weak and strong solutions of this SDE.

A missing reference is added

References in corpus (1)

• Markovian bridges: Weak continuity and pathwise constructions